We are tasked with finding the minimum and maximum values of the function $f(x) = \frac{x^2 + 1}{x}$ on the interval $[-2, 0)$.

Step 1: Rewrite the Function

First, simplify $f(x)$:

$f(x) = \frac{x^2 + 1}{x} = x + \frac{1}{x}$

Step 2: Compute the Derivative

To find the critical points, we first need to compute the derivative of $f(x)$.

$f'(x) = \frac{d}{dx}\left( x + \frac{1}{x} \right) = 1 - \frac{1}{x^2}$

Step 3: Solve for Critical Points

Set $f'(x) = 0$ to find the critical points:

$1 - \frac{1}{x^2} = 0$

$\frac{1}{x^2} = 1 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = \pm 1$

However, since $x$ is restricted to the interval $[-2, 0)$, only $x = -1$ is valid.

Step 4: Evaluate the Function at Critical Points and Endpoints

Now, evaluate $f(x)$ at the critical point and at the endpoints of the interval.

• At $x = -1$:

$f(-1) = -1 + \frac{1}{-1} = -1 - 1 = -2$

• At $x = -2$:

$f(-2) = -2 + \frac{1}{-2} = -2 - \frac{1}{2} = -\frac{5}{2}$

• At $x \to 0^-$ (as $x$ approaches 0 from the left):

$f(x) = x + \frac{1}{x} \quad \text{approaches} \quad -\infty$ as $\frac{1}{x}$ tends to $-\infty$.

Step 5: Determine Minimum and Maximum Values

From the evaluations:

• The maximum value is $f(-1) = -2$.
• The function decreases without bound as $x \to 0^-$, so the minimum value does not exist (the function tends to $-\infty$).

Conclusion:

• The maximum value of $f(x)$ on $[-2, 0)$ is $-2$.
• The function has no minimum value as it tends to $-\infty$ as $x \to 0^-$.

Would you like further details on this solution?

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Here are 5 related questions:

1. How would the problem change if the interval included 0?
2. What is the behavior of $f(x)$ on a different interval, such as $(-\infty, 0)$?
3. Can you explain the role of critical points in determining extremum values?
4. What are the conditions for a function to have absolute extrema?
5. How does asymptotic behavior influence the analysis of functions near boundary points?

Tip: Always analyze the behavior of functions near the boundaries when dealing with open intervals to understand how the function behaves as it approaches those points.